Decision Theory Matters for Financial Advice

Abstract

We show that the optimal asset allocation for an investor depends crucially on the decision theory with which the investor is modeled. For the same market data and the same client data different theories lead to different portfolios. The market data we consider is standard asset allocation data. The client data is determined by a standard risk profiling question and the theories we apply are mean–variance analysis, expected utility analysis and cumulative prospect theory. For testing the robustness of our results, we carry out the comparisons for alternative data sets and also for variants of the risk profiling question.

Acknowledgements

Appendix

Summary statistics for the monthly-returns data set; the statistics is for annualized data

GSCITR

HFRIFFM

I3M

JPMBD

MSEM

MXWO

NAREIT

PE

Min.

\(-\)3.7205

\(-\)2.3395

\(-\)0.0885

\(-\)0.4539

\(-\)5.1791

\(-\)2.1162

\(-\)3.0722

\(-\)3.1710

Mean

0.0441

0.0584

0.0149

0.0364

0.0669

0.0669

0.0546

0.0594

Median

0.0909

0.1208

0.0065

0.0303

0.0872

0.1259

0.1402

0.1283

Max.

2.3879

1.3723

0.1839

0.5459

3.8823

1.6683

2.2549

3.2807

SD

0.8279

0.4503

0.0762

0.1732

1.2644

0.5335

0.6409

0.7517

Skewness

\(-\)0.4903

\(-\)1.8369

0.4415

0.0302

\(-\)0.5283

\(-\)0.6643

\(-\)0.8683

\(-\)0.2963

Kurtosis

4.5874

10.604

2.1578

3.5438

4.7248

4.8595

7.6889

8.0357

Table 4

The empirical distribution, constructed via k-means clustering (\(k=15\)) from the monthly returns data set with the last column corresponding to the appended European call-option on MXWO

P

GSCITR

HFRIFFM

I3M

JPMBD

MSEM

MXWO

NAREIT

PE

Call on MXWO

0.1731

0.0441

0.0168

0.0012

0.0060

0.0484

0.0268

0.0315

0.0191

0

0.1635

\(-\)0.0024

0.0041

0.0023

0.0062

\(-\)0.0157

0.0023

0.0107

0.0109

0

0.1490

\(-\)0.0384

0.0213

0.0002

\(-\)0.0012

0.0938

0.0318

0.0205

0.0271

0

0.1346

0.0756

0.0098

0.0009

0.0030

\(-\)0.0175

\(-\)0.0095

\(-\)0.0213

\(-\)0.0044

0

0.1010

\(-\)0.0816

\(-\)0.0025

0.0007

\(-\)0.0023

\(-\)0.0294

\(-\)0.0082

\(-\)0.0052

\(-\)0.0174

0

0.0865

0.0654

0.0350

\(-\)0.0023

0.0056

0.1353

0.0500

0.04918

0.0682

0

0.0865

0.0014

\(-\)0.0261

0.0060

0.0008

\(-\)0.1334

\(-\)0.0518

\(-\)0.0401

\(-\)0.0691

0

0.0385

\(-\)0.0284

\(-\)0.0117

0.0051

0.0023

\(-\)0.1663

\(-\)0.0074

\(-\)0.0354

0.0013

0

0.0241

\(-\)0.1388

\(-\)0.0376

\(-\)0.0029

\(-\)0.0038

\(-\)0.1643

\(-\)0.1059

\(-\)0.1447

\(-\)0.1658

0

0.0096

0.0370

0.0634

\(-\)0.0019

\(-\)0.0024

0.1929

0.0840

0.0978

0.2623

0

0.0096

\(-\)0.0574

\(-\)0.0743

\(-\)0.0026

0.0266

0.1505

0.0933

0.1499

0.0562

0

0.0096

0.1482

0.0240

\(-\)0.0058

0.0168

0.3165

0.1232

0.1467

0.1585

0.2832

0.0048

\(-\)0.3100

\(-\)0.1935

0.0064

\(-\)0.0113

\(-\)0.3817

\(-\)0.1763

\(-\)0.2560

\(-\)0.2643

0

0.0048

\(-\)0.1459

\(-\)0.1950

0.0080

\(-\)0.0035

\(-\)0.2370

\(-\)0.1050

\(-\)0.0862

\(-\)0.1193

0

0.0048

\(-\)0.0314

\(-\)0.1313

0.0005

0.0252

\(-\)0.4316

\(-\)0.1259

\(-\)0.0833

\(-\)0.1161

0

Table 5

The CPT parameter values from Abdellaoui et al. (2007), the computed \(\gamma \) and \(\delta \) values, the computed losses \(-x\) in the lottery as well as the corresponding \(\kappa \), \(\alpha \) and \(\theta \) parameters in the alternative objective functions

\(\alpha ^+\)

\(\alpha ^-\)

\(\beta \)

\(\gamma \)

\(\delta \)

loss: \(-x\) (lottery)

\(\kappa \) (quad util.)

\(\alpha \) (MV)

\(\theta \) (CRRA)

S1

1.03

0.7

4.11

0.57

0.86

\(-\)0.01

3.14

4.43

\(-\)46.8

S2

1.20

1.24

1.08

0.61

0.64

\(-\)0.20

\(-\)0.61

\(-\)0.6

2.22

S3

1.02

0.60

2.25

0.71

0.71

\(-\)0.01

2.99

4.08

\(-\)27.5

S4

0.85

0.80

2.75

0.65

0.68

\(-\)0.04

2.28

2.72

\(-\)8.34

S5

0.57

0.85

0.85

1.00

0.58

\(-\)0.34

\(-\)1.20

\(-\)1.41

3.59

S6

0.56

1.01

2.34

0.89

0.58

\(-\)0.15

\(-\)0.10

\(-\)0.10

1.21

S7

0.96

0.79

1.85

0.65

0.58

\(-\)0.07

1.63

1.79

\(-\)3.90

S8

2.17

1.09

6.80

0.64

0.64

\(-\)0.01

3.09

4.33

\(-\)38.6

S9

1.28

0.66

2.08

0.53

0.89

\(-\)0.01

3.09

4.32

\(-\)38.3

S10

1.02

0.70

1.28

0.54

0.71

\(-\)0.05

2.06

2.38

\(-\)6.47

S11

0.72

0.68

1.72

0.91

0.65

\(-\)0.06

1.75

1.94

\(-\)4.47

S12

0.74

0.91

1.20

0.61

0.57

\(-\)0.21

\(-\)0.67

\(-\)0.69

2.34

S13

0.72

0.67

1.39

0.68

0.68

\(-\)0.08

1.29

1.37

\(-\)2.48

S14

1.93

3.07

0.30

0.50

0.53

\(-\)0.60

\(-\)1.11

\(-\)1.69

4.17

S15

0.59

0.82

1.53

0.71

0.71

\(-\)0.14

0.19

0.19

0.59

S16

0.61

0.82

2.01

0.81

0.60

\(-\)0.11

0.64

0.65

\(-\)0.48

S17

1.00

0.85

3.06

0.56

0.52

\(-\)0.04

2.23

2.64

\(-\)7.84

S18

0.68

0.82

1.63

0.86

0.81

\(-\)0.10

0.88

0.90

\(-\)1.13

S19

0.96

0.80

1.54

0.62

0.60

\(-\)0.08

1.27

1.34

\(-\)2.39

S20

0.90

0.71

7.01

0.81

0.75

\(-\)0.01

3.18

4.55

\(-\)60.3

S21

0.87

1.02

1.93

0.71

0.63

\(-\)0.12

0.39

0.39

0.14

S22

0.69

0.66

1.13

0.75

0.68

\(-\)0.12

0.55

0.55

\(-\)0.23

S23

0.84

0.55

1.70

0.75

0.68

\(-\)0.03

2.71

3.48

\(-\)15.0

S24

0.50

0.54

1.57

0.61

0.63

\(-\)0.06

1.83

2.05

\(-\)4.94

S25

0.61

0.33

2.53

0.67

0.63

\(-\)0.01

3.29

4.86

\(-\)221

S26

1.11

0.75

2.94

0.78

0.83

\(-\)0.02

2.90

3.87

\(-\)21.6

S27

0.38

0.48

0.87

0.48

0.97

\(-\)0.09

1.15

1.21

\(-\)1.99

S28

0.51

0.51

4.35

0.64

0.85

\(-\)0.01

3.22

4.65

\(-\)80.7

S29

0.87

0.75

2.06

0.63

0.63

\(-\)0.05

2.02

2.31

\(-\)6.12

S30

1.24

0.66

0.63

0.63

0.86

\(-\)0.08

1.33

1.41

\(-\)2.60

S31

2.05

0.71

1.92

0.68

0.71

\(-\)0.01

3.20

4.59

\(-\)66.4

S32

0.49

1.01

0.41

0.91

0.61

\(-\)0.89

\(-\)0.89

\(-\)1.59

4.23

S33

0.72

1.08

1.46

0.68

0.58

\(-\)0.23

\(-\)0.83

\(-\)0.86

2.65

S34

0.46

0.80

1.06

0.61

0.65

\(-\)0.25

\(-\)0.97

\(-\)1.05

2.97

S35

0.71

0.60

3.77

0.96

0.64

\(-\)0.01

3.05

4.22

\(-\)33.0

S36

0.56

0.58

2.05

0.86

0.73

\(-\)0.04

2.44

2.98

\(-\)10.1

S37

0.86

0.81

4.32

0.59

0.56

\(-\)0.03

2.63

3.33

\(-\)13.3

S38

1.32

0.74

1.57

0.64

0.91

\(-\)0.03

2.70

3.45

\(-\)14.7

S39

0.66

0.60

1.76

0.55

0.60

\(-\)0.05

2.14

2.50

\(-\)7.09

S40

0.71

0.59

1.68

0.86

0.83

\(-\)0.04

2.46

3.02

\(-\)10.9

S41

0.50

0.51

0.87

0.58

0.64

\(-\)0.15

\(-\)0.09

\(-\)0.09

1.19

S42

0.69

0.66

1.14

0.58

0.59

\(-\)0.12

0.43

0.43

0.05

S43

0.67

0.81

1.66

0.65

0.68

\(-\)0.11

0.71

0.73

\(-\)0.67

S44

0.60

0.62

2.41

0.65

0.62

\(-\)0.04

2.40

2.91

\(-\)9.65

S45

0.51

1.57

0.40

0.54

0.91

\(-\)0.76

\(-\)0.98

\(-\)1.65

4.22

S46

0.63

0.50

2.37

0.52

0.55

\(-\)0.02

2.93

3.94

\(-\)23.3

S47

0.66

0.87

0.80

0.81

1.00

\(-\)0.25

\(-\)0.94

\(-\)1.00

2.88

S48

1.35

0.43

1.64

0.68

0.63

\(-\)0.003

3.27

4.80

\(-\)149

Table 6

Original data set: distances between the portfolios measured in terms of CE, expressed in percents and showing annualized values